\(\int \frac {x^2 (4+x^3)}{(1+x^3)^{3/4} (1+x^3+x^4)} \, dx\) [1183]

Optimal result

Integrand size = 28, antiderivative size = 87 \[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=-\sqrt {2} \arctan \left (\frac {-\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^3}}{\sqrt {2}}}{x \sqrt [4]{1+x^3}}\right )-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right ) \]

[Out]

-2^(1/2)*arctan((-1/2*2^(1/2)*x^2+1/2*(x^3+1)^(1/2)*2^(1/2))/x/(x^3+1)^(1/4))-2^(1/2)*arctanh(2^(1/2)*x*(x^3+1
)^(1/4)/(x^2+(x^3+1)^(1/2)))

Rubi [F]

\[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=\int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx \]

[In]

Int[(x^2*(4 + x^3))/((1 + x^3)^(3/4)*(1 + x^3 + x^4)),x]

[Out]

-(x*Hypergeometric2F1[1/3, 3/4, 4/3, -x^3]) + (x^2*Hypergeometric2F1[2/3, 3/4, 5/3, -x^3])/2 + Defer[Int][1/((
1 + x^3)^(3/4)*(1 + x^3 + x^4)), x] - Defer[Int][x/((1 + x^3)^(3/4)*(1 + x^3 + x^4)), x] + 4*Defer[Int][x^2/((
1 + x^3)^(3/4)*(1 + x^3 + x^4)), x] + Defer[Int][x^3/((1 + x^3)^(3/4)*(1 + x^3 + x^4)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {1}{\left (1+x^3\right )^{3/4}}+\frac {x}{\left (1+x^3\right )^{3/4}}+\frac {1-x+4 x^2+x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}\right ) \, dx \\ & = -\int \frac {1}{\left (1+x^3\right )^{3/4}} \, dx+\int \frac {x}{\left (1+x^3\right )^{3/4}} \, dx+\int \frac {1-x+4 x^2+x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx \\ & = -x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {4}{3},-x^3\right )+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{4},\frac {5}{3},-x^3\right )+\int \left (\frac {1}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}-\frac {x}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}+\frac {4 x^2}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}+\frac {x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )}\right ) \, dx \\ & = -x \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {4}{3},-x^3\right )+\frac {1}{2} x^2 \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{4},\frac {5}{3},-x^3\right )+4 \int \frac {x^2}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx+\int \frac {1}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx-\int \frac {x}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx+\int \frac {x^3}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.85 \[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=-\sqrt {2} \left (\arctan \left (\frac {-x^2+\sqrt {1+x^3}}{\sqrt {2} x \sqrt [4]{1+x^3}}\right )+\text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{1+x^3}}{x^2+\sqrt {1+x^3}}\right )\right ) \]

[In]

Integrate[(x^2*(4 + x^3))/((1 + x^3)^(3/4)*(1 + x^3 + x^4)),x]

[Out]

-(Sqrt[2]*(ArcTan[(-x^2 + Sqrt[1 + x^3])/(Sqrt[2]*x*(1 + x^3)^(1/4))] + ArcTanh[(Sqrt[2]*x*(1 + x^3)^(1/4))/(x
^2 + Sqrt[1 + x^3])]))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.84 (sec) , antiderivative size = 208, normalized size of antiderivative = 2.39

method	result	size
trager	\(\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \left (x^{3}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}-2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}+1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (-\frac {2 \sqrt {x^{3}+1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}+2 \left (x^{3}+1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}-2 \left (x^{3}+1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}+1}\right )\)	\(208\)

[In]

int(x^2*(x^3+4)/(x^3+1)^(3/4)/(x^4+x^3+1),x,method=_RETURNVERBOSE)

[Out]

RootOf(_Z^4+1)*ln(-(RootOf(_Z^4+1)^3*x^4-RootOf(_Z^4+1)^3*x^3-2*(x^3+1)^(1/4)*RootOf(_Z^4+1)^2*x^3+2*(x^3+1)^(
1/2)*RootOf(_Z^4+1)*x^2-2*(x^3+1)^(3/4)*x-RootOf(_Z^4+1)^3)/(x^4+x^3+1))+RootOf(_Z^4+1)^3*ln(-(2*(x^3+1)^(1/2)
*RootOf(_Z^4+1)^3*x^2+2*(x^3+1)^(1/4)*RootOf(_Z^4+1)^2*x^3+RootOf(_Z^4+1)*x^4-RootOf(_Z^4+1)*x^3-2*(x^3+1)^(3/
4)*x-RootOf(_Z^4+1))/(x^4+x^3+1))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.26 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.21 \[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (\frac {\left (i + 1\right ) \, \sqrt {2} x + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (\frac {-\left (i - 1\right ) \, \sqrt {2} x + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) - \left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \log \left (\frac {\left (i - 1\right ) \, \sqrt {2} x + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) + \left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \log \left (\frac {-\left (i + 1\right ) \, \sqrt {2} x + 2 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

integrate(x^2*(x^3+4)/(x^3+1)^(3/4)/(x^4+x^3+1),x, algorithm="fricas")

[Out]

-(1/2*I + 1/2)*sqrt(2)*log(((I + 1)*sqrt(2)*x + 2*(x^3 + 1)^(1/4))/x) + (1/2*I - 1/2)*sqrt(2)*log((-(I - 1)*sq
rt(2)*x + 2*(x^3 + 1)^(1/4))/x) - (1/2*I - 1/2)*sqrt(2)*log(((I - 1)*sqrt(2)*x + 2*(x^3 + 1)^(1/4))/x) + (1/2*
I + 1/2)*sqrt(2)*log((-(I + 1)*sqrt(2)*x + 2*(x^3 + 1)^(1/4))/x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate(x**2*(x**3+4)/(x**3+1)**(3/4)/(x**4+x**3+1),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{3} + 4\right )} x^{2}}{{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate(x^2*(x^3+4)/(x^3+1)^(3/4)/(x^4+x^3+1),x, algorithm="maxima")

[Out]

integrate((x^3 + 4)*x^2/((x^4 + x^3 + 1)*(x^3 + 1)^(3/4)), x)

Giac [F]

\[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=\int { \frac {{\left (x^{3} + 4\right )} x^{2}}{{\left (x^{4} + x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {3}{4}}} \,d x } \]

[In]

integrate(x^2*(x^3+4)/(x^3+1)^(3/4)/(x^4+x^3+1),x, algorithm="giac")

[Out]

integrate((x^3 + 4)*x^2/((x^4 + x^3 + 1)*(x^3 + 1)^(3/4)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2 \left (4+x^3\right )}{\left (1+x^3\right )^{3/4} \left (1+x^3+x^4\right )} \, dx=\int \frac {x^2\,\left (x^3+4\right )}{{\left (x^3+1\right )}^{3/4}\,\left (x^4+x^3+1\right )} \,d x \]

[In]

int((x^2*(x^3 + 4))/((x^3 + 1)^(3/4)*(x^3 + x^4 + 1)),x)

[Out]

int((x^2*(x^3 + 4))/((x^3 + 1)^(3/4)*(x^3 + x^4 + 1)), x)